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Weighted Context-Free-Language Ordered Binary Decision Diagrams

Meghana Sistla, Swarat Chaudhuri, Thomas Reps

TL;DR

The paper addresses the challenge of compactly representing high-arity, semi-field-valued functions in quantum-simulation contexts by introducing Weighted CFLOBDDs (WCFLOBDDs), which fuse the weighted edge framework of WBDDs with the procedure-call, hierarchical structure of CFLOBDDs. The core idea is a level-based hierarchy that halves variables at each level, with level-0 transitions carrying weights and a CFL-based matched-path constraint ensuring a CFL-language interpretation. The authors prove canonicity via structural invariants and a formal Canonicity Theorem, and they provide algorithms for common operations (e.g., matrix and vector operations) within this framework. Empirical results on synthetic and quantum-circuit benchmarks show that WCFLOBDDs often outperform both WBDDs and CFLOBDDs in terms of problem size that can be handled within a fixed time, supporting the claim that WCFLOBDDs offer a practical best-of-both-worlds approach for certain quantum-simulation tasks, albeit with nuanced runtime behavior.

Abstract

This paper presents a new data structure, called \emph{Weighted Context-Free-Language Ordered BDDs} (WCFLOBDDs), which are a hierarchically structured decision diagram, akin to Weighted BDDs (WBDDs) enhanced with a procedure-call mechanism. For some functions, WCFLOBDDs are exponentially more succinct than WBDDs. They are potentially beneficial for representing functions of type $\mathbb{B}^n \rightarrow D$, when a function's image $V \subseteq D$ has many different values. We apply WCFLOBDDs in quantum-circuit simulation, and find that they perform better than WBDDs on certain benchmarks. With a 15-minute timeout, the number of qubits that can be handled by WCFLOBDDs is 1-64$\times$ that of WBDDs (and 1-128$\times$ that of CFLOBDDs, which are an unweighted version of WCFLOBDDs). These results support the conclusion that for this application -- from the standpoint of problem size, measured as the number of qubits -- WCFLOBDDs provide the best of both worlds: performance roughly matches whichever of WBDDs and CFLOBDDs is better. (From the standpoint of running time, the results are more nuanced.)

Weighted Context-Free-Language Ordered Binary Decision Diagrams

TL;DR

The paper addresses the challenge of compactly representing high-arity, semi-field-valued functions in quantum-simulation contexts by introducing Weighted CFLOBDDs (WCFLOBDDs), which fuse the weighted edge framework of WBDDs with the procedure-call, hierarchical structure of CFLOBDDs. The core idea is a level-based hierarchy that halves variables at each level, with level-0 transitions carrying weights and a CFL-based matched-path constraint ensuring a CFL-language interpretation. The authors prove canonicity via structural invariants and a formal Canonicity Theorem, and they provide algorithms for common operations (e.g., matrix and vector operations) within this framework. Empirical results on synthetic and quantum-circuit benchmarks show that WCFLOBDDs often outperform both WBDDs and CFLOBDDs in terms of problem size that can be handled within a fixed time, supporting the claim that WCFLOBDDs offer a practical best-of-both-worlds approach for certain quantum-simulation tasks, albeit with nuanced runtime behavior.

Abstract

This paper presents a new data structure, called \emph{Weighted Context-Free-Language Ordered BDDs} (WCFLOBDDs), which are a hierarchically structured decision diagram, akin to Weighted BDDs (WBDDs) enhanced with a procedure-call mechanism. For some functions, WCFLOBDDs are exponentially more succinct than WBDDs. They are potentially beneficial for representing functions of type , when a function's image has many different values. We apply WCFLOBDDs in quantum-circuit simulation, and find that they perform better than WBDDs on certain benchmarks. With a 15-minute timeout, the number of qubits that can be handled by WCFLOBDDs is 1-64 that of WBDDs (and 1-128 that of CFLOBDDs, which are an unweighted version of WCFLOBDDs). These results support the conclusion that for this application -- from the standpoint of problem size, measured as the number of qubits -- WCFLOBDDs provide the best of both worlds: performance roughly matches whichever of WBDDs and CFLOBDDs is better. (From the standpoint of running time, the results are more nuanced.)
Paper Structure (10 sections, 1 theorem, 4 equations, 4 figures)

This paper contains 10 sections, 1 theorem, 4 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Background
  3. Weighted Binary Decision Diagrams (WBDDs)
  4. Semi-Field Weights
  5. CFLOBDDs as Hierarchical Finite-State Machines
  6. Weighted CFLOBDDs (WCFLOBDDs)
  7. Basic Structure of WCFLOBDDs
  8. Canonicity
  9. Structural Invariants.
  10. Canonicity Theorem

Key Result

theorem 1

If $C_1$ and $C_2$ are two level-$k$ WCFLOBDDs for the same Boolean function, and use the same variable ordering, then $C_1$ and $C_2$ are isomorphic.

Figures (4)

  • Figure 1: The design space of BDDs, WBDDs, CFLOBDDs, and WCFLOBDDs.
  • Figure 2: WBDDs for the matrices $H_2$, $H_4$, and $H_8$, with $x$ variables for rows and $y$ variables for columns.
  • Figure 3: (a), (b), and (c) show WCFLOBDDs for the first three matrices in $\mathcal{H}$---$H_2$, $H_4$, and $H_8$---with the variable ordering $\langle x_0, y_0, x_1, y_1, \ldots \rangle$ ($\vec{x}$: row; $\vec{y}$: column). (d) shows the general structure of a WCFLOBDD that represents $H_{2^i}$. (e) illustrates the constituents of the WCFLOBDD for $H_2$.
  • Figure 4: Object diagram of the WCFLOBDD for matrix $H_2$ (Fig. \ref{['Fi:wcflobdd_hadamard']}(a)).

Theorems & Definitions (7)

  • definition 1
  • definition 2: Mock-WCFLOBDD; see Fig. \ref{['Fi:wcflobdd_hadamard']}(e)
  • definition 3: Mock-proto-WCFLOBDD
  • definition 4: Exit vertex reached via assignment $a$
  • definition 5: WCFLOBDD evaluation
  • definition 6
  • theorem 1